Unique factorization domain

This article gives a basic definition in the following area: ring theory
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A unique factorization domain is an integral domain in which any non-zero element $x$ can be written as a product $x=up_{1}\dots p_{n}$, where $u$ is a unit and the $p_{i}$s are irreducible elements uniquely, that is, if $x=u'q_{1}\dots q_{m}$, with the $q_{i}$s irreducible, then $m=n$ and there is some permutation $\sigma \in S_{n}$ such that $p_{i}=q_{\sigma (i)}$ for all $i$.

Relation to other properties

Stronger properties

• Principal ideal domain
• Euclidean domain
• Field

Weaker properties

• Integral domain
• Ring

Examples

• $\mathbb {Z}$ is a unique factorization domain - this is the Fundamental Theorem of Arithmetic.
• The Gaussian integers $\mathbb {Z} [i]$.

Results

• The ring of integers of a number field is a unique factorization domain (also, a principal ideal domain) if and only if its ideal class group is the trivial group.